| L(s) = 1 | + 8·3-s + 104·7-s − 158·9-s − 320·11-s + 100·13-s − 580·17-s − 720·19-s + 832·21-s + 1.68e3·23-s − 1.09e3·27-s + 108·29-s − 9.84e3·31-s − 2.56e3·33-s − 6.54e3·37-s + 800·39-s − 1.06e4·41-s + 2.56e4·43-s + 2.82e4·47-s − 2.52e4·49-s − 4.64e3·51-s − 3.13e4·53-s − 5.76e3·57-s − 3.08e4·59-s + 2.45e4·61-s − 1.64e4·63-s + 3.45e4·67-s + 1.35e4·69-s + ⋯ |
| L(s) = 1 | + 0.513·3-s + 0.802·7-s − 0.650·9-s − 0.797·11-s + 0.164·13-s − 0.486·17-s − 0.457·19-s + 0.411·21-s + 0.665·23-s − 0.289·27-s + 0.0238·29-s − 1.83·31-s − 0.409·33-s − 0.785·37-s + 0.0842·39-s − 0.986·41-s + 2.11·43-s + 1.86·47-s − 1.50·49-s − 0.249·51-s − 1.53·53-s − 0.234·57-s − 1.15·59-s + 0.844·61-s − 0.521·63-s + 0.941·67-s + 0.341·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 8 T + 74 p T^{2} - 8 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 104 T + 36038 T^{2} - 104 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 320 T + 319702 T^{2} + 320 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 100 T + 297086 T^{2} - 100 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 580 T + 2475814 T^{2} + 580 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 720 T + 4633798 T^{2} + 720 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1688 T + 10950502 T^{2} - 1688 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 108 T + 39233214 T^{2} - 108 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9840 T + 76732702 T^{2} + 9840 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6540 T + 61572814 T^{2} + 6540 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10620 T + 77106582 T^{2} + 10620 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 25672 T + 364912302 T^{2} - 25672 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 28296 T + 617782998 T^{2} - 28296 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 31340 T + 755347886 T^{2} + 31340 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 30800 T + 1666896598 T^{2} + 30800 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 24540 T + 1447225822 T^{2} - 24540 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 34584 T + 2930656478 T^{2} - 34584 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12400 T + 1143670702 T^{2} - 12400 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7180 T + 284279286 T^{2} - 7180 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 71840 T + 7430807198 T^{2} + 71840 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 31928 T + 5910993662 T^{2} + 31928 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 40748 T + 8570866774 T^{2} + 40748 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 190140 T + 19653817414 T^{2} - 190140 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.150715654758133503119119812100, −8.928644299927674612867761385159, −8.297947411345426001551412807626, −8.265416168700353898850054263644, −7.53117292956262574402920429100, −7.36204504990993838540248179529, −6.76782670639879189501212821775, −6.22170399017713411089829720323, −5.63042350703467306300018648136, −5.38628822684066071775240159189, −4.76545121818945487991171063289, −4.44290511773649865377901547823, −3.64368547294366013148469142315, −3.36775349567206882953586468810, −2.51890780897631140804481378902, −2.37873136269807428299156163690, −1.61913159847714433186143971861, −1.10618291727522451464225124557, 0, 0,
1.10618291727522451464225124557, 1.61913159847714433186143971861, 2.37873136269807428299156163690, 2.51890780897631140804481378902, 3.36775349567206882953586468810, 3.64368547294366013148469142315, 4.44290511773649865377901547823, 4.76545121818945487991171063289, 5.38628822684066071775240159189, 5.63042350703467306300018648136, 6.22170399017713411089829720323, 6.76782670639879189501212821775, 7.36204504990993838540248179529, 7.53117292956262574402920429100, 8.265416168700353898850054263644, 8.297947411345426001551412807626, 8.928644299927674612867761385159, 9.150715654758133503119119812100
